In this chapter, we prove the well-posedness in \(L^2\) -based Sobolev spaces of the mixed boundary value and mixed-transmission problems for the compressible-framework anisotropic Stokes system in a bounded Lipschitz domain of \({\mathbb R}^n\) , \(n \geq 2\) , with an internal Lipschitz interface that intersects transversally the boundary of the domain. The (viscosity) coefficient tensor is assumed to have entries in \(L^{\infty }\) and is required to satisfy the relaxed ellipticity and symmetry conditions. The main results are the existence and uniqueness of a weak solution for nonlinear mixed and mixed-transmission problems for the compressible-framework anisotropic Navier-Stokes system with small data in \(L^2\) -based Sobolev spaces in a bounded Lipschitz domain with the same geometry as in the linear case, but for \(n = 2\) or \(n = 3\) . Moreover, we obtain an existence result for the mixed problem for a semi-linear anisotropic, incompressible Stokes type system with arbitrary data in \(L^2\) -based Sobolev spaces on a bounded Lipschitz domain in \({\mathbb R}^n\) , for \(n = 2\) or \(n = 3\) , by using the Leray-Schauder fixed point Theorem.